Social contagions on multiplex networks with different reliability
Introduction
The diffusion of information, rumor, innovation, epidemic can be modeled as spreading dynamics on complex networks, which have attracted much attention in recent years [[1], [2], [3], [4], [5], [6]]. According to different contents, researchers divided the spreading dynamics into simple and complex contagions [7]. For the simple contagion, the probability that a susceptible node infected by two different neighbors are independent [[2], [8], [9]]. However, for the complex contagions (i.e., social contagions), a node’s decision in adopting the contagion is dependent on neighbors’ attitude, thus the social reinforcement effect is included [[10], [11], [12], [13], [14], [15], [16], [17]]. The inherent social reinforcement character makes that the spreading process, final size, and phase transition are markedly affected [18]. For instance, Watts found that the heterogeneity of degree distribution makes the system less vulnerable to global contagions [15]. Zhu et al. revealed that the heterogeneities of degree and weight distributions observably affect the behavior of social contagions [19].
The multiplexity of networks markedly affect cascading failures, spreading dynamics, synchronization, and game [[20], [21], [22], [23], [24], [25], [26]]. Many studies have focused on the effects of network structures on social contagions, such as degree correlations, clustering, multiplexity, and temporal networks [[27], [28], [29], [30], [31], [32], [33]]. Brummitt and his colleagues [31] proposed a generalized Markovian Watts threshold model [15] on multiplex networks. They assumed that a susceptible node becomes active only when the fraction of active neighbors is larger than the threshold in any layer. Through the percolation theory and extensive numerical simulations, they found that the multiplexity of networks facilitates the contagions. Recently, Wang et al. [29] proposed a non-Markovian behavior spreading model which takes the communication channel alteration mechanism into account, and they developed a generalized edge-based compartmental theory to describe the model. They found the communication channel alteration character suppresses the contagions, and in ER–SF multiplex networks introduces a crossover phenomenon in adoption size from continuous to discontinuous.
For a contagion process, different communication channels (i.e., different layer) have distinct reliability [34]. For the diffusion of high-tech products in society, an individual receiving a piece of information from specialist increases more reliability than that from an individual who lacks the basic education. Thus, we should consider the effects of distinct reliability of layers. Yaǧan and Gligor [34] proposed a Markovian Watts threshold model on multiplex networks, and assumed that each layer has different reliability. Their analysis indicates a directed subgraph on vulnerable nodes. However, the effects of different reliability on non-Markovian social contagions have not been investigated systematically.
In this paper, we propose a non-Markovian social contagion model on multiplex networks, and assume that each layer has different reliability. We develop a generalized edge-based compartmental theory to describe the model. From theory and numerical simulations we find that increasing the reliability of different layers promotes the social contagions, but does not alter the growth pattern of the final contagion size versus contagion information probability.
In Section 2, we describe our social contagion model on multiplex networks. In Section 3, we detail our edge-based compartmental theory. In Section 4, we present numerical simulation results. In Section 5, we present conclusions and discussions.
Section snippets
Model descriptions
To model social contagions on multiplex networks, we consider for simplicity, two layers and with the same number of nodes . Each node in layer is matched one-to-one with that in layer , which means that nodes in different communication channels. Denoting node ’s duplicates in layers and as and respectively, and the two duplicates are coupled nodes of each other. Thus, node and its two duplicates have the same states in two different layers. The two layer networks can be
Theory
To quantificationally describe the social contagions, we develop a generalized edge-based compartmental theory. In theory, we assume that the multiplex network is large, sparse and local tree-like, and the spreading dynamics evolves continuously. From the descriptions of the model in Section 2, the non-Markovian character is included, and a randomly selected node and its two duplicates have the same states. Since the strong dynamical correlations among the states of neighbors induced by the
Numerical simulations
We preform extensive numerical simulations on artificial networks on both ER–ER and SF–SF coupled multiplex networks. For the ER–ER multiplex networks, both layers are ER networks with degree distributions and , where and are the average degrees of layers and respectively. For the SF–SF multiplex networks, both layers are scale-free networks with power-law degree distributions and , where ,
Discussions
In reality, different communication channel always carries distinct reliability. However, its effect on the dynamics of non-Markovian social contagions is still neglected. In this work, we first proposed a non-Markovian social contagion model on multiplex networks. It is assumed that each susceptible node becoming adopted only when its received accumulated reliability in two layer is larger than its adoption threshold. To describe this proposed non-Markovian model, a generalized edge-based
Acknowledgment
This work was supported by the Basic and Frontier Research Program of Chongqing, China (Grant No. cstc2015jcyjA40025).
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